I have made my first working database application. On my computer it works without any problems, but when I install/copy it to somebody elses computer in exactly the same path it can't/won't open the database.

I've installed the BDE on the other computers (it comes with the install program). But do I have to make an alias or something on the other computer? What do I do wrong?

You must install BDE in other PC and design your Table alias name as a directory ( for example: Table1.databasename:= c:\myprogram\ ). Naturally you have to put your table into this directory.
Delphi database applications doesn't work without BDE.

In the BDE on the other PC (the one without the database), you must add an alias to point to the location of the database in the format of:
f:\myprog\tables -> where tables is the directory where the tables reside, plus, you must map a drive on the pc where the application is being run to drive F:
OR
in the BDE on the other PC (the one without the database), point it to the location of the tables using UNC, eg:
\\MyDBServer\MyProg\Tables, then no drive mapping is required.